On the eigenfunctions of the Douglas-Kroll operator

The matrix elements of the quasirelativistic Douglas-Kroll operators up to the fourth order for hydrogen-like ions is constructed with as few additional approximations as possible, to investigate the behaviour of its 1s eigenfunctions in the vicinity of the nucleus. Because Douglas-Kroll is a momentum space theory, we use a basis set of spherical waves which are eigenfuntions of the square of the momentum operator. While this avoids the most serious approximation of the standard Douglas-Kroll-Hess protocol, namely that the basis functions used to construct the Douglas-Kroll operator are eigenfunctions of the (squared) momentum operator, it also makes the convergence of this expansion very slow, because spherical waves are not well suited to represent the (weak) singularities of the eigenfunctions at the position of the point-like nucleus. On the other hand, the convergence is quite monotonic, and information on the behaviour close to the nucleus can be extracted from the convergence rate. Starting with the second-order, the eigenfunctions of the Douglas-Kroll operator are not “more singular” than the Dirac eigenfunctions, and the occurence of an additional error when using regular basis sets, as postulated in the literature, can not be observed. The resolution of the identity, which is involved in any practical approach to construct the matrix elements of the Douglas-Kroll operators beyond the first order, is a minor problem for heavy nuclei.